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The Pearson product-moment correlation coefficient ( written as r for sample estimate,  for parameter )  Z a Z b n-1 i = 1 n r = Where z a and z b are z scores for each person on some measure a and some measure b, and n is the number of people

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The following method of inverting a matrix is taken largely from: Swokowski, Earl W. (1979) Fundamentals of College Algebra. Boston, MA: Prindle, Weber & Scmidt

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The first step is to form a matrix which has the same number of rows as the original correlation matrix of predictors, but has twice as many columns. The original predictor correlations are placed in the left half, and an equal order identity matrix is place in the right half: (Predictor correlations) (Identity matrix) (Swokowski, Earl W. (1979) Fundamentals of College Algebra. Boston, MA: Prindle, Weber & Scmidt)

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Though a series of calculations called elementary row transformations, the goal is to change all the numbers in the matrix so that the identity matrix is on the left, and a new matrix is on the right: Identity Matrix Inverse Matrix (Swokowski, Earl W. (1979) Fundamentals of College Algebra. Boston, MA: Prindle, Weber & Scmidt)

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Swokowski, Earl W. (1979) Fundamentals of College Algebra. Boston, MA: Prindle, Weber & Scmidt “ MATRIX ROW TRANSFORMATION THEOREM Given a matrix of a system of linear equations, each of the following transformations results in a matrix of an equivalent system of linear equations: (i) (i)Interchanging any two rows (ii) (ii)Multiplying all of the elements in a row by the same nonzero real number k. (iii) (iii)Adding to the elements in a row k times the corresponding elements of any other row, where k is any real number. “

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OUR CALCULATIONS VALUES FROM SPSS          The difference has to do with rounding error. There are so many transformations in matrix math that all computations must be carried out with many, many significant figures, because the errors accumulate. I only used what was visible in my calculator. Good matrix software should use much more precision. This is a relatively brief equation to solve. Imagine the error that can accumulate with hundreds of matrix transformations. This is a very important point, and one should always be certain the software is using the appropriate degree of precision.,